chua_english ver

1. Random Number Generator Based on a Chaotic Dynamical System

The aim of research is to obtain a
pseudo-random binary sequence
using a hardware and software
implementations of Chua’s circuit.
Oleg Maksimovich Opyakin1,
Konstantin Dmitrievich Lishik1,
Daniil Alexeevich Vikultsev1
Scientific advisor: Stanislav Vladilenovich Vinogradov1
1Moscow Institute of Physics and Technology
(national research institute)

2. Plan of Research

1.
2.
3.
4.
5.
Objectives
Setup & mathematical model
Hardware implementation
Obtaining random numbers
Processing & results
2

3. SETUP & MATHEMATICAL MODEL

SETUP & MATHEMATICAL MODEL
3

4. Chua’s diode

Voltage-current characteristics
of Chua’s diode
Elecrtonic scheme
of Chua’s diode
4

5. Chua’s Circuit

• Chaotic oscillations:
– Voltages on C1 and C2
– Current through L
5

6. Mathematical model

- Kirchhoff law
- dimensionless coefficients
- function of Chua’s diode
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7. Positions of equilibrium

Positions of equilibrium
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8. Phase portraits

Unstable “3-D focuses”
Stable “3-D focus”
Finally:
Unstable 2-D focus
• Positions of equilibrium of experimental setup will match E1, E2 and E3
• The system will evolve from E1 and E2, towards E3
Kuznetsov N. “Scenario of the Birth of Hidden Attractors in the Chua Circuit”
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9. Numerical solution

Euler’s method implemented with Python:
• Positions of equilibrium match E1, E2 and E3
• The system evolves from E1 and E2
• Unclear behavior near E3
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10. HARDWARE IMPLEMENTATION

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11. Measuring setup

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12. Hardware implementation

A double-scroll attractor
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13. OBTAINING RANDOM NUMBERS

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14. Obtaining random numbers

State vector:
x > 0 – the right state
x < 0 – the left state
- characteristic time of the system
In our system:
= 5 ms
= 10
We will monitor the system states
.
Hypothesis:
Probabilities of finding the dynamic system in the left or right states are equal
Thus:
Right state - 0
Left state - 1
Binary sequence
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15. RESULTS & PROCESSING

RESULTS & PROCESSING
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16. Sequence visualization

Example:
100101011
1
0
0
1
0
1
0
1
1
NO visible patterns
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17. NIST tests

Experimental data
Experimental satistics
Probabity theory
Etalon satistics
Comparing experimental and
etalon statistics
P - value
Probability that generator generates true random numbers
P > 0.01
the sequence is random
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18. NIST tests results

All of P – values are more than 0.01, thus, the generated sequence is random
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19. Results

1.
2.
The behavior of the system has been described mathematically and
studied numerically and experimentally. Theoretical assumptions match
with the experiment.
Random numbers can be obtained by the developed method.
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20. Literature

[1] L. Chua, “The genesis of Chua’s sircuit”, 1992
[2] D. V. Sivukhin, “Electricity” 6th Edition, FITMAZLIT, Moscow, 2019.
[3] V.I. Arnold, “Ordinary Differential Equations”, 4 th edition, Izhevsk, 2000
[4] Y.S. Ilyashenko, “Attractors of Dynamic Systems”, 2008
[5] A.S. Dmitriev, “Chaos generators”, Moscow
[6] NIST, “A Statistical Test Suite for Random and Pseudorandom Number
Generators for Cryptographic Applications ”, 2010
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21.

ADDITIONAL SLIDES
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22. Verification with the law of iterated logarithm

Law of iterated logarithm:
Example:
1
0
0
1
0
1
-1
-1
1
-1
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23. Equilibrium positions

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24. NIST test example

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